An Indian Olympiad Math Question | Can you solve ?

追加: 2026-05-06

[00:00:01]Hello, welcome back once again.

[00:00:03]Today we have this interesting functional equation f of x plus f of 1 divided by 1 minus x is equal to x plus 1.

[00:00:13]In this video, we're going to find the value for f of 2. So let's get started.

[00:00:19]Now, whenever we given this type of question when we asked to compute f of 2, then we're going to make the following first substitution that is letting x equals 2.

[00:00:32]Then from the original equation, we get f of 2 then plus. So here, 1 minus 2 here is negative 1. So 1 divided by negative 1 will give us negative 1. So here, we now have f of negative 1.

[00:00:48]So this is equal to from the right-hand side, 2 plus 1 giving us 3.

[00:00:53]So what should be our next substitution?

[00:00:56]So our next substitution will come from this equation. Now, we are looking for f of 2, but here we have f of negative 1 involved. So we need another equation that we also have f of negative 1. So our next substitution tells us from here that we're going to let x equals negative 1. Now from the original equation, we see from here now, we have f of negative 1 then plus. So here, 1 minus minus 1 is 1 plus 1 which is 2. So we have here f of 1 over 2. So this is equal to x plus 1 where x is equal to negative 1 will give us 0.

[00:01:41]So are we done? No. We need f of 1 over 2 in another third equation.

[00:01:47]You can see assuming we have f of 3 here, then we can solve both equations simultaneously. But here, we have another another term here involving the f of 1 over 2. So our next substitution is letting x equals 1 over 2.

[00:02:08]So from the original equation, we now have f of 1 over 2 plus. So if you check the second term there, so f of 1 divided by 1 minus 1 over 2 if you check properly, that is going to give us f of 2.

[00:02:23]So this is equal to 1 over 2 plus 1 will give us 3 over 2.

[00:02:28]Now, this we have three equations, so we need to solve them for f of 2, right? Now, let us look for where f of 2 and f of negative 1 is is involved in the second equation, and I don't think we have anything like that, right?

[00:02:47]>> [clears throat] >> Okay, we don't need to subtract this first equation from this one. When you do that, you will eliminate f of 2.

[00:02:54]Meanwhile, we are looking for f of 2. So what are we going to do? Let us subtract Is that what we're going to do?

[00:03:06]Should we subtract this equation from the second one here?

[00:03:12]Okay, I think what to do here is to subtract this second equation from this third one, right? So because if you do that, we get f of 1 over 2 eliminated.

[00:03:21]So here, f of 2 minus f of negative 1 is what we're going to have on the left-hand side. So we have f of 2 minus f of negative 1 is equal to so 3 over 2 minus 0 is 3 over 2.

[00:03:37]You know why we have only these two on the left-hand side because f of 1 over 2 minus f of 1 over 2 is equal to 0. Now we can see that this equation 1 and this one, they are similar. We have the same terms, right? So if what happens when you add both equations? f of negative 1 will be eliminated, right? So let's add the two equations. So bring down the equation that is f of 2 plus f of negative 1 is equal to 3. So add both equations.

[00:04:08]Once you add the equations, f of negative 1 will be eliminated. We now have 2 f of 2 from here is equal to so 3 over 2 plus 3. So I think that is going to give us 9 over 2, right?

[00:04:27]Yes, so that is going to give us 9 over 2. So from here, we multiply both sides by 1 over 2, right? So 1 over 2 times 2 f of 2 is equal to 1 over 2 times 9 over 2. So these both get canceled. Then we have our solution that f of 2 is equal to 9 over 4. And this right over here is our solution. Thank you for watching. If you enjoyed the video, please kindly subscribe to this channel.

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